The function $g(x) = (x - 3)^3$ can be described as a transformation of the function $f(x) = x^4$, though the two are different types of polynomial functions: one is cubic, the other quartic. Let's break down the transformation conceptually.

General Comparison:

1. Base Function Types:

   • $f(x) = x^4$ is a quartic function (degree 4), which has a different shape compared to $g(x) = (x - 3)^3$, a cubic function (degree 3).
   • Quartic functions typically have a "wider" parabolic-like shape, while cubic functions have an "S" shape with inflection points.

2. Transformation of Function:

   • While $g(x)$ and $f(x)$ are different types of functions, one transformation concept can still apply: horizontal translation.

3. Horizontal Shift:

   • The graph of $g(x) = (x - 3)^3$ is a shift of the standard cubic function $f(x) = x^3$, not $f(x) = x^4$.
   • Specifically, the transformation is a shift 3 units to the right. This comes from the $x - 3$ inside the parentheses. The graph of $f(x) = x^3$ is moved so that the inflection point at $(0, 0)$ is now at $(3, 0)$.

Conclusion:

The graph of $g(x) = (x - 3)^3$ is best described as a rightward horizontal shift of the cubic function $f(x) = x^3$, by 3 units. It doesn't relate directly to the quartic function $f(x) = x^4$, since their shapes and degrees are different.

Do you want to dive deeper into any specific part of this transformation?

Here are 5 related questions to consider:

1. How do horizontal and vertical transformations differ for polynomial functions?
2. What is the shape of the graph of $f(x) = x^3$ and how do its key features change when transformed?
3. What does the degree of a polynomial function say about its general shape and behavior?
4. How does a horizontal shift affect the inflection point of a cubic function?
5. How would additional transformations, like vertical stretching or reflection, affect the graph of a cubic function?

Tip: When transforming functions, always focus on how changes to the equation affect the shape, position, and orientation of the graph.